Publication: Time-Parallel Computation of Pseudo-Adjoints for a Leapfrog Scheme
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Time-Parallel Computation of Pseudo-Adjoints for a Leapfrog Scheme

- Article in a journal -
 

Area
Oceanography

Author(s)
C. H. Bischof , H. M. Bücker , P. T. Wu

Published in
International Journal of High Speed Computing

Year
2004

Abstract
The leapfrog scheme is a commonly used second-order difference scheme for solving differential equations. If Z(t) denotes the state of a system at a particular time step t, the leapfrog scheme computes the state at the next time step as Z(t+1) = H(Z(t), Z(t-1), W), where H is the nonlinear timestepping operator and W represents parameters that are not time-dependent. In this note, we show how the associativity of the chain rule of differential calculus can be used to compute a so-called adjoint, the derivative of a scalar-valued function applied to the final state Z(T) with respect to some chosen parameters, efficiently in a parallel fashion. To this end, we (1) employ the reverse mode of automatic differentiation at the outermost level, (2) use a sparsity-exploiting version of the forward mode of automatic differentiation to compute derivatives of H at every time step, and (3) exploit chain rule associativity to compute derivatives at individual time steps in parallel. We report on experimental results with a 2-D shallow water equations model problem on an IBM SP parallel computer and a network of Sun SPARCstations.

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BibTeX
@ARTICLE{
         Bischof2004TPC,
       author = "C. H. Bischof and H. M. B{\"u}cker and P.T. Wu",
       title = "Time-Parallel Computation of Pseudo-Adjoints for a Leapfrog Scheme",
       journal = "International Journal of High Speed Computing",
       pages = "1--27",
       doi = "doi:10.1142/S0129053304000219",
       abstract = "The leapfrog scheme is a commonly used second-order difference scheme for solving
         differential equations. If~$Z(t)$ denotes the state of a system at a particular time step~$t$, the
         leapfrog scheme computes the state at the next time step as~$Z({t+1}) = H(Z(t), Z({t-1}), W)$,
         where~$H$ is the nonlinear timestepping operator and~$W$ represents parameters that are not
         time-dependent. In this note, we show how the associativity of the chain rule of differential
         calculus can be used to compute a so-called adjoint, the derivative of a scalar-valued function
         applied to the final state~$Z(T)$ with respect to some chosen parameters, efficiently in a parallel
         fashion. To this end, we (1)~employ the reverse mode of automatic differentiation at the outermost
         level, (2)~use a sparsity-exploiting version of the forward mode of automatic differentiation to
         compute derivatives of~$H$ at every time step, and (3)~exploit chain rule associativity to compute
         derivatives at individual time steps in parallel. We report on experimental results with a 2-D
         shallow water equations model problem on an IBM~SP parallel computer and a network of Sun
         SPARCstations.",
       ad_area = "Oceanography",
       ad_tools = "Adifor",
       ad_theotech = "Sparsity",
       year = "2004",
       volume = "12",
       number = "1"
}


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